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Robotics Research
l^hnical Report







On the Existence of Generic Coordinates

by
Thomas Dubd



Technical Report No. 456

Robotics Report No. 203

June, 1989




\,



I"*

Im

02



CO

e
o

a EH
o

>j D
2 Q



O

Hterm((/.,(/i)) > i^+[P .
D

Notation: For / an ideal of A' and P G PP,, let Np{I,d) denote the
number of distinct head terms of the form x^~'^x^^iP which belong to poly-
nomials in /. That is,

Np{I,d) = |{c : x^^x:^iPGHead(/)}|

= \{c : x^=x:^ieHead(/):P}| .

Though it will not be used in this report, one might note that that is the
Hilbert function ,{fi), . . ■ ,(f>i{fm) must also be linearly indepen-
dent. Triangulating the 4>,{fj), it is possible to create linear combinations
gi,...,gm '^ J oi the form

m

m

; = i

such that a, J € K{y), and the head terms of the 9, are distinct. Then
Qi — ,{h,) where h, = J2T=i ^'.ifj- Each h, has a head term of the form
xf~'^x^^.iP, and thus by the preceding lemma, Hterm(g,) is therefore also
of this form. And so each of the gi, ■ . ■ ,gm ^ J has a distinct head term of
the form xf~'^x^^iP and therefore Np{J,d) > c.

A symmetric argument employing the inverse homomorphism (f>J^ shows
that Np{I',d) > Np{J,d), completing the proof of the lemma.
D

Lemma 4 Let I £ A, and I' be the ideal generated by I in A'. Suppose
that I' contains a polynomial h with Hterm((;/i, (/i)) = i^'^x^^jP. Then
Np{I,d) >c + l.

Proof. Since Hterm((^,(/i)) = i^-'x^+iP,

d

,{h) = J^a,x'l-'x',^,P + L.O.T.

«=c

Using the inverse transformation (pj^, h can be written as:

d

h = (^a.x^-(x,,i-yxJ')P + L.O.T.

- i:«.(E(-ir'( • )y'-x^v,^,)p+ L.O.T.



3 THE CHANGE OF COORDINATES 7

The a, are elements of K{y) and hence can be written as a, = Y^T=m^i,2y'-
Collecting ternis around powers of y

^ = Ej/^(EI:^-.+;(-1)-M • ]x''r^xU,)P + h.O.T.

z=m ,-c ;=0 \ -' /

00 d d ( \

The polynomial h has been written in the form k = J^V^^z- From the
properties of polynomial rings with extended coefficient fields, it follows
that each k^ G /. Let r = min{2|3, such that 6,_2_, / 0}. Then, for
q = r - d,...,r,

K = i:x^V,^,(y:(-in( ;. )6.,,_.>,)P + L.0.T.

If y < r — 9 then q — i + j;)P + L.0.T.

The largest power product remaining in the expression for /i, is x^'^'^'^x^^^P,
and it's coefficient is

Hcoef(/t,) = E(-ir^^' (,!,)''«,-



1 = C



= E(-i)-'-L" A.



where /?, denotes the value 6, ,.- 1- Consider the /?, to be variables. There
are (f — c + 1 of these variables:



3 THE CHANGE OF COORDINATES 8

Now the d+ 1 expressions Hcoef(/ir_d), . • • ,Hcoef(/ir) are linearly indepen-
dent over these variables. Therefore, fixing the values of any d — c + I of
the Hcoef(/ifc) has the effect of fixing the values of the /9,'s. Therefore, if
d — c + 1 of these expressions evaluate to zero, then all of the /?, must be
zero. This contradicts the choice of r as the least value v^'here at least one
of the 0, is non-zero. Therefore, at least c+ 1 of the Hcoef(/ii) are non-zero,
and Head(7) contains at least c + I different power products of the form

D

Corollary 5 If Np{4>,{r),d) = c, then x'^^-^4+iP ^ Head(


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Online LibraryThomas DubeOn the existence of generic coordinates → online text (page 1 of 1)